where we have written \(\Delta ^{\textrm {o}}=\Delta \setminus \partial \Delta \)  for the interior or bulk of \(\Delta \) . Under appropriate gauge-fixing, this amplitude can be shown to depend only on the boundary data and on the topology of \(\Delta \)  and to never depend on the details of the bulk cellular complex \(\Delta ^{\textrm {o}}\)  [1]}, [2]}, [3]}, [4]}.

The sum over bulk spins and bulk intertwiners is usually divergent, just as Feynman diagrams in quantum field theory. It is possible to render those amplitudes finite by \(q\) -deforming \(\mathrm {SU}(2)\)  at root of unity, with \(q=\exp (2i\pi /N+2)\)  for an integer \(N\in {\mathbb {N}}\) . This gives the Turaev-Viro topological invariant [5]} and is interpreted as switching on a non-vanishing positive cosmological constant \(\Lambda >0\) . Even without quantum deforming the gauge group, one can still identify the translational gauge symmetry responsible for the divergences and gauge-fix them - typically by fixing the value of the spins on the edges belonging to a maximal tree in \(\Delta ^{\textrm {o}}\) , in which case the gauge-fixed amplitudes never depend neither on the choice of the gauge-fixing tree nor on the bulk cellular complex \(\Delta ^{\textrm {o}}\)  [1]}, [7]}, [8]}, [9]}.

More precisely, the amplitude for a 3D cellular complex with the topology of a 3-ball remains the evaluation of its boundary spin network. For 3D cellular complexes with a non-trivial topology, one needs to evaluate the spin network wave-function on the values of the non-trivial holonomies along the non-contractible cycles of the bulk cellular complex or integrate over possible values [2]}, [3]}, [4]}.
